Roughly chop the kimchi into smaller pieces. Cover half of each tortilla with shredded cheddar cheese, then add the chopped kimchi on top.
Heat a large skillet over medium, then add the butter or oil. Fry or scramble the eggs in the butter or oil until they reach your desired doneness.
Transfer the cooked eggs to the tortillas, placing them on top of the kimchi. Fold the empty side of the tortilla over the filled side to close.
Transfer the folded quesadillas back to the same skillet used to cook the eggs and continue to cook over medium heat until the cheese is melted and the tortilla is browned and crispy on each side.
Slice the cooked kimchi quesadillas in half and enjoy!